Most detailed explanation in the blogosphere is "Why Stuff is Hard" by Jim Pivarski,

http://cornellmath.wordpress.com/2007/08/08/why-stuff-is-har...

http://cornellmath.wordpress.com/2007/08/11/why-everything-i...

http://cornellmath.wordpress.com/2007/08/11/why-everything-i...

Complete argument is the Dyson, Lieb and Thirring's proof of stability of matter, http://projecteuclid.org/euclid.bams/1183555452 .

Many thanks for that last link.

But if one atom "tries" to pass through another, the electrons from both atoms will be spatially very close to each other, and consequently, they will be occupying very similar states. Coulombic repulsion certainly also plays a role, but it's not the only force at work here. I don't know the relative magnitudes for single atoms bumping into each other, but for bulk material, I believe it's about 40% exclusion, 60% coulomb.

Why would electrons in different atoms interact differently than electrons in the same atom? They don't, as far as I know. Furthermore, if electrons from different atoms only interacted Coulombically (repelling each other), molecules and solids would never get formed.

These are difficult questions, not addressed in the clear-cut cases often presented in introductory QM textbooks. I can't think of any must-read references right now, but if you're interested in this kind of thing I know of two books on molecular quantum mechanics; one by McWeeney (methods of molecular quantum mechanics) and one by Atkins (molecular quantum mechanics), which I haven't read myself, but I hear it's good. I don't know of any free online sources for this kind of stuff, but it's probably out there, I just haven't really looked yet.

Not just electrons. Take a look at Fermions: http://en.wikipedia.org/wiki/Fermion